## Is the product of normals normal?

You can use moments to see that the product XY of independent normals cannot be normal except in trivial cases. By trivial, I mean V(X)V(Y)=0.

## Is the product of two normal variables normal?

The product of two normal PDFs is proportional to a normal PDF. Note that the product of two normal random variables is not normal, but the product of their PDFs is proportional to the PDF of another normal.

Is the product of normal random variables normal?

The distribution of the product of normal variables is not, in general, a normally distributed variable. However, under some conditions, is showed that the distribution of the product can be approximated by means of a Normal distribution.

What happens if two independent normal random variables are combined?

This means that the sum of two independent normally distributed random variables is normal, with its mean being the sum of the two means, and its variance being the sum of the two variances (i.e., the square of the standard deviation is the sum of the squares of the standard deviations).

### Is product of gaussians a Gaussian?

It is well known that the product and the convolution of Gaussian probability density functions (PDFs) are also Gaussian functions.

### Is the product of two lognormal distributions lognormal?

The distribution of the product of two random variables which have lognormal distributions is again lognormal. This is itself a special case of a more general set of results where the logarithm of the product can be written as the sum of the logarithms.

Is the sum of two dependent normal random variables normal?

If and form a bivariate normal distribution, then their sum is normal.

Is the difference between two normal distributions normal?

The difference is not even necessarily normally distributed if the 2 normal random variables are not bivariate normal, which can happen if they are not independent.. In addition to the assumption pointed out by Mark, you are also ignoring the fact that the means are different.

#### What is the expectation of XY?

– The expectation of the product of X and Y is the product of the individual expectations: E(XY ) = E(X)E(Y ). More generally, this product formula holds for any expectation of a function X times a function of Y . For example, E(X2Y 3) = E(X2)E(Y 3).